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Find the general indefinite integral.

$ \displaystyle \int 2^t (1 + 5^t)\, dt $

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00:57

Frank Lin

00:50

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Missouri State University

Oregon State University

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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01:19

right. The theme in this problem is that if you are asked to take a to the T. Power D. T. So the location of your expense important, uh The anti griddle is one divided by the natural log of T. Um times A. To the T. Power. And it's just undoing the chamber and you do need to remember plus your constant. So the key in this problem then is to get your problem to look like that uh to the T. One plus five to the T. Power DT. So what we need to do is because there's a multiplication right here, we need to distribute that in there before trying anything. And a rule here is that because we have to the T. Power. So there's T. Number of twos and there's T. Number of fives. That should be the same as two times 52 times five tee times. So it's the same thing as 10 to the T. Power. I hope my explanation made sense why that's true because now we can just use this rule up here. Uh put a boxer on that, that's what we're using. Um And it's going to be won over natural log of two times, two to the T. Power, plus one over natural log of 10 times 10 to the T. Power. And don't forget about plus C. You can verify that this is correct because the derivative of this will get us back to here, which we already know is equal to that, so we have the correct answer.

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